Today I’m going to revisit a topic that a month ago I committed to stop writing about — the productivity-income quagmire. Neoclassical economists argue that income is proportional to productivity. The problem is that they have no way of measuring productivity that is independent of income. So in practice, they test their theory by assuming it’s true. They show that one form of income — value added per worker — is correlated with another form of income — wages.

In No Productivity Does Not Explain Income, I pointed out this circularity. And I haven’t heard the end of it since. The problem, critics say, is that my argument is flawed. Yes, value added and wages are related by an accounting definition. But that doesn’t ‘guarantee’ (in strict mathematical terms) a correlation between the two. In an update post, I admitted that the critics were correct. It is *possible* for wages and value added per worker to be uncorrelated.

Does this mean my argument is fatally flawed? No. But I will admit that the language in the original post was slightly overheated. The accounting definition between wages and value added doesn’t *guarantee* a correlation between the two. It *virtually* guarantees one.

I’m going to show you here that yes, it’s possible for wages and value added per worker to be uncorrelated. But it’s so unlikely that it’s not worth considering. The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker.

### The accounting definition

Let’s revisit how neoclassical economists test their theory of income. What they *should* do is measure the productivity of workers and see how it relates to income. But for a host of reasons, economists don’t do this. The most basic problem is that workers do different activities. This makes it impossible to compare their outputs. How, for instance, do you compare the output of a musician with the output of a farmer? The only logical answer is that you *can’t* — at least not objectively.

Realizing this problem should have caused neoclassical economists concern. It means that their theory is untestable. But instead of admitting defeat, neoclassical economists devised a loophole. To test their theory, they assume that one form of income (value added) actually measures *productivity*. Then they compare this income to another form of income (wages). The two types of income are correlated! Productivity explains income!

No.

The problem is that economists never actually measure productivity. Instead, they show that two types of income are correlated. But these two types of income are related by an accounting definition. And this accounting definition, I claim, (virtually) guarantees a correlation.

Let’s dive into the math.

Economists want to show that wages correlate with productivity — a fundamental tenet of their theory of income. To do this, they measure productivity in terms of value added per labor hour (or sales per labor hour, if data for value added is unavailable).

Let’s put this test in math form. Let *w* be the average wage of workers in a firm. Let *Y* be the value added of the firm. And let *L* be the labor hours worked by all workers in the firm. Economists look for a correlation between average wages (*w*) and value-added per labor hour (*Y/L*):

The problem is that the left-hand and right-hand sides of this comparison aren’t independent from each other. Instead, they’re connected by an accounting definition. Let’s look at it.

Value added is, by definition, the sum of the wage bill (*W*) and profits (*P*):

Now let’s calculate value added per labor hour. We take value added *Y* and divide it by labor hours *L*. To maintain the equality, we also divide the right-hand side by *L*:

Now let’s distribute the *L* to the two terms in the numerator:

What is *W / L*? It’s the wage bill divided by the number of labor hours. This happens to be the average hourly wage, *w*. So value-added per worker is equivalent to the average wage plus some noise term:

Given this accounting definition, it’s unsurprising that wages correlate with value added per labor hour. But do the two terms *have* to correlate? In strict math terms, no. If the noise term is large, it will drown out the correlation between wages and value added.

The problem is that the noise term is itself likely to correlate with value added per worker. Why? Because the noise term is actually profit per labor hour (*P/L*). It shares the same denominator (*L)* with value add per labor hour (*Y / L*). This codependency increases the chance of correlation.

So yes, it is possible for wages and value added to be uncorrelated. But it is also extremely improbable.

### Throwing numbers into the accounting formula

I want to show you just how probable it is that we’ll find a correlation between wages and value added per worker. Here’s how I’m going to do it. I’m going to randomly generate values for the wage bill (*W*), profit (*P*) and labor hours (*L*) of imaginary firms. Then I’ll throw these values into the accounting definition. Finally, I’ll look for a correlation between the average wage and value added per labor hour.

Just so I’m clear, here’s the steps:

- Draw random values for the wage bill
*W*and profit*P*of firms - Use these numbers to define value added:
*Y = W + P* - Randomly generate a value for the number of labor hours (
*L*) worked in each firm - Compare value added per labor hour (
*Y/L*) to the average wage (*w = W/L*)

To generate the random numbers, we have to assume some sort of distribution. I’ll assume that *W*, *P* and *L* come from a lognormal distribution — a distribution that is common among economic phenomena. Because I want a general test, I’ll also assume that the parameters of this lognormal distribution are themselves random. See the notes for the math details and code.

If you want a simple metaphor for this process, think of defining *W*, *P* and *L* by rolling a dice. But each time we roll, the number of dice varies. This means that *W*, *P* and *L* can vary over a large range.

### Judging correlation

With our random numbers in hand, we plug them into our accounting definition and look for a correlation between value added per worker and the average wage. Here things get slightly more complicated.

It’s easy to measure correlation — just calculate the correlation coefficient (*r*) or the coefficient of determination (*R ^{2}*). The problem is that correlation exists on a scale. To say that two variables are ‘correlated’, we need to decide on some arbitrary threshold. This entails a value judgment.

For better or for worse (mostly for worse), the standard practice in econometrics is to hide this value judgment in more math. We define something called the ‘*p*-value’ of the correlation. Then we judge correlation (or lack thereof) based on an arbitrary threshold in the *p*-value.

What’s a ‘*p*-value’? It’s a probability. The *p*-value tells you the probability of getting your observed correlation (or greater) from random numbers. The lower the *p*-value, the more ‘statistically significant’ your correlation.

Using *p*-values depends on a host of assumptions, many of which are violated when we study economic phenomena. Worse still, there are many ways to rig the game so you get better (i.e. lower) *p*-values. It’s called *p*-hacking, and it’s a huge problem in the social sciences.

Despite these problems, I’m going to use *p*-values to judge the correlation between simulated wages and value added per worker. I do so not because I like *p*-values (I don’t), but because using them is the standard practice in econometrics. Getting a low *p*-value means your results are publishable. Your correlation is ‘significant’! (If you want to read about how silly this is, check out the book The cult of statistical significance.)

### Correlation from randomness

With our *p*-values in tow, here’s the question we want to ask. If we plug random numbers into our accounting definition, how likely is it that the correlation between wages and value added per worker will be ‘statistically significant’?

The answer, it seems, is *very likely*. Although we’re dealing with random numbers, a ‘significant’ correlation between simulated wages and value added per worker seems to be in the cards. Table 1 tells the story.

**Table 1: Correlation from randomness**

P-value (%) | Portion of results below p-value (%) |
---|---|

5.00 | 99.96 |

1.00 | 99.94 |

0.10 | 99.91 |

0.01 | 99.89 |

Let’s unpack the results. I have a model that throws random numbers into our accounting definition. For each set of random numbers, I calculate the *p*-value of the correlation between wages and value added per labor hour. Then I look at how often these *p*-values are below some critical value. The left-hand column in Table 1 shows various thresholds for the *p*-value. The right-hand column shows the portion of the simulations in which the *p*-value is below this critical value for ‘statistical significance’.

By throwing random numbers into our accounting definition, we get a ‘statistically significant’ correlation 99.9% of the time. Note that this holds no matter how stringent our level of statistical significance. Even for the very low (in the social sciences) *p*-value of 0.0001, some 99.89% of the results are ‘statistically significant’. It seems that by throwing numbers into our accounting definition, a ‘statistically significant’ correlation between wages and value added per worker is virtually guaranteed.

Let’s look at the simulation results another way. Figure 1 shows how the *p*-values are distributed across all of the simulations. Most of the *p*-values are so small (meaning the correlation is so ‘significant’) that I have to plot *p* on a log scale. The *x*-axis shows the logarithm of the *p*-value. The *y*-axis shows the relative frequency of each *p*-value. To get some perspective, the vertical red line shows the standard threshold for statistical significance, a *p*-value of 0.05. Virtually all of the results are below this value, meaning the correlation is ‘statistically significant’.

Figure 2 shows yet another way of looking at the simulation results. Instead of plotting *p*-values, here I look at the distribution of the *f*-statistic. The *f*-statistic is another way of measuring ‘statistical significance’ (*p*-values are actually derived from the *f*-statistic). But whereas a *lower* *p*-value is ‘more significant’, a *higher* *f*-statistic is ‘more significant’.

As with *p*-values, the *f*-statistics are so extreme that I need to plot their logarithm. The vertical red line in Figure 2 shows the threshold for statistical significance at the 5% level. Results with an *f*-statistics above this value are deemed ‘statistically significant’. Again, we see that the vast majority of results are ‘statistically significant’.

It seems that by throwing random numbers into our accounting definition, we can’t help but find a correlation between wages and value added per worker.

### Manufacturing correlation

The charge that I’ve laid against neoclassical economists is that when they test their theory of income, they’re fooling themselves. Their method (virtually) guarantees a positive result. They regress two types of income — wages and value added per labor hour — that are related by an accounting definition. The problem is that this accounting definition (virtually) guarantees a statistically significant correlation.

Now the degree to which this correlation is guaranteed depends on the specifics of how the wage bill, profit and labor hours are distributed. But what I’ve shown here is that across a huge class of numbers, a ‘statistically significant’ correlation is almost unavoidable. It’s manufactured by our accounting definition.

This result highlights a problem with how economists use *p*-values. The use of *p*-values has turned into a production function: run a regression → get a low *p*-value → get published. Rarely do economists question whether the assumptions behind *p*-values are actually met in the real-world data.

In my simulation, the assumptions behind *p*-values are systematically violated. To use them, we must assume that our data is ‘statistically independent’. Here, this means that wages and value added per labor hour can be treated as independent, random variables. The problem is that they’re not independent. Wages and value add per labor hour are related by an accounting definition. This renders them highly *dependent*. So the use of *p*-values is moot.

Still, *p*-values are the standard by which economists judge correlation. By this standard, our accounting definition virtually guarantees a ‘statistically significant’ correlation between wages and value added per labor hour.

### The larger problem

The larger problem here is that the marginal productivity theory of income is untestable. Its core components — productivity and the ‘quantity’ of capital — cannot be measured objectively. If you want to know more about these problems, I recommend John Pullen’s book The Marginal Productivity Theory of Distribution: A Critical History.

The reality is that when it comes to explaining income, there is a long and sordid history of political economists fooling themselves. Neoclassical economists may be the most visible fools, but they’re by no means the only ones. Marxists too test their theory of income in circular terms. If you’re interested in Marxist theory, check out the debate between Jonathan Nitzan, Shimshon Bichler and the Marxist Paul Cockshott. What appears to be evidence for the labor theory of value, Nitzan and Bichler show, is actually mathematical foolery.

This issue of circular testing cuts to a core problem in both neoclassical and Marxist theories of income. Both explain income in terms of quantities that are unobservable. Unsurprisingly, tests of these theories resort to circular logic. Such tests invariably show that two forms of income are correlated. Then they claim that one form of income is something other than what it seems.

Sadly, this foolery has been standard practice for a century. And that’s not really surprising. There is perhaps no topic in which objectivity is more difficult than the distribution of income. Still, if we want a scientific theory of income, we need to do better. We need to stop fooling ourselves.

### Notes

Here’s my model. I assume that profit (*P*), the wage bill (*W*) and the number of labor hours (*L*) in firms are random variables that are lognormally distributed. If you’re not familiar, the lognormal distribution looks like a bell curve when you take the logarithm of its values. If the variable *x* is lognormally distributed, log(*x*) is normally distributed. Many quantities in economics are lognormally distributed, which is why I use this function here.

The lognormal distribution has two parameters, the ‘location’ parameter mu and the ‘scale’ parameter sigma. I’ll denote the lognormal distribution with the notation used in R. If *x* is lognormally distributed with parameters mu and sigma, I denote it as:

`x = lnorm(mu, sigma)`

To assume almost nothing about the distribution of *P*, *W* and *L*, I let the parameters of the lognormal distribution vary randomly over a uniform distribution. I’ll denote the uniform distribution using the notation used in R. If *x* is a uniformly distributed over the range 0 to 1, we write:

`x = runif(0, 1)`

I let the parameters of the lognormal distribution vary between 0 and 10. If you’re familiar with the lognormal distribution, you’ll know that this is a huge parameter space. So the values of *P*, *W* and *L* are:

`P = lnorm( mu = runif(0, 10), sigma = runif(0, 10))`

`W = lnorm( mu = runif(0, 10), sigma = runif(0, 10))`

`L = lnorm( mu = runif(0, 10), sigma = runif(0, 10))`

I take these values and throw them into the accounting definition. Average wages are then *W / L*. Value added per worker is *(W + P) / L*. To see how the correlation between wages and value added per worker varies, I run the algorithm several thousand times.

### Code

Here’s the R code for the model. Run it for yourself and see what you find.

library(doSNOW) n_test = 20000 n_firms = 10^4# clustercl = makeCluster(4, type="SOCK") registerDoSNOW(cl) clusterSetupRNG (cl, type = "RNGstream")# progress Barpb = txtProgressBar(max = n_test, style = 3) progress = function(n) setTxtProgressBar(pb, n) opts = list(progress = progress) test = foreach(i = 1:n_test, .options.snow=opts, .combine=rbind) %dopar% {# wagebillmu = runif(1, 0, 10) sigma = runif(1, 0, 10) wagebill = rlnorm(n_firms, mu, sigma)# profitmu = runif(1, 0, 10) sigma = runif(1, 0, 10) profit = rlnorm(n_firms, mu, sigma)# value added (sum of wagebill and profit)value_added = wagebill + profit# labor hoursmu = runif(1, 0, 10) sigma = runif(1, 0, 10) labor_hours = rlnorm(n_firms, mu, sigma)# hourly wagehourly_wage = wagebill / labor_hours# valued added per labor hourva_per_hour = value_added / labor_hours# regress value added per hour and hourly wager = lm( log(va_per_hour) ~ log(hourly_wage) ) f = summary(r)$fstatistic f_stat = f[1] p = pf(f[1],f[2],f[3],lower.tail=F) r2 = summary(r)$r.squared output = data.frame(f_stat, p, r2) } stopCluster(cl)# portion that are significantsig_frac = length(test$p[ test$p < 0.05 ]) / length(test$p)# f statistichist(log10(test$f_stat), breaks = 100, xlim = c(0, 10)) abline(v = log10(3.85), col = "red" )# exportwrite.csv(test, "test_data.csv")

### Further reading

Cockshot, P., Shimshon, B., & Nitzan, J. (2010). Testing the labour theory of value: An exchange. http://bnarchives.yorku.ca/308/02/20101200_cockshott_nitzan_bichler_testing_the_ltv_exchange_web.htm

Pullen, J. (2009). *The marginal productivity theory of distribution: A critical history*. London: Routledge.

Ziliak, S., & McCloskey, D. N. (2008). *The cult of statistical significance: How the standard error costs us jobs, justice, and lives*. University of Michigan Press.

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You spend a lot of time attacking theory instead of assumptions, but it is the assumptions that tell you everything about what economists seek to abstract into oblivion.

The problem here is the assumption that “the iron clad law of supply and demand” sets prices. That’s empirically false for anything other than commodities. In most situations, prices are set by the sellers based on how much profit they want to make. In this sense, wages fully reflect the value provided by labor, from the perspective of the owner of the business. Ironically, then, the insistence that wages reflect the actual value addded by labor is pretty much an admission that the “iron clad law of supply and demand” is a dead letter.

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Hi Tao,

I agree that the most important way to criticize a theory is to show that it’s assumptions are flawed. This is true of the core of a theory. But it’s also true of tests of a theory.

What I’m doing here is criticizing how economists test marginal productivity theory. They claim that the correlation between wages and value added per worker is evidence for their theory. But this is based on flawed assumptions.

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[…] [3] There are many other blows to human capital theory that are fatal. Most importantly, the theory posits that productivity explains income. But economists never measure productivity independently of income. Instead, all of their evidence for productivity is, in fact, circularly tied to income. For details, see these posts: No, Productivity Does Not Explain Income, Productivity Does Not Explain Wages, Debunking the ‘Productivity-Pay Gap’, and Productivity and Income … Again. […]

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I’m an engineer so I understand your math and there’s nothing wrong with your theory.

But there is another theory that looks correct to me.

Most people and companies operate in a competitive market. I got 3 bids to paint my house. I chose the one I thought was the best value. I considered price and quality.

The owner of this painting company will go broke unless he hires productive painters and pays them as much as they could make elsewhere. He competes for my business and he competes for his labor supply. He cannot take money from anyone unfairly.

In a competitive market, people will be paid approximately according to their productivity.

How can we compare a musician to a farmer? In a competitive market, millions of people will bid for their services, and the price will be approximately correct.

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You’re arguing by assumption. To ‘prove’ your point (that people earn their marginal product) you’re assuming that prices are ‘correct’. And by ‘correct’, you mean that prices reveal productivity. It’s a circular operation. You can’t assume what you want to prove.

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I do not understand why you keep going at it. Your weird “neo classics are dumb as a rock and missed the most obvious flaw” is just… it’s just weird.

1. The simulation you run is just silly:

The question you raise boils down to this “is the sum of two **independent and identically distributed** random variables correlated with each of these random variables”. There is no need for a simulation, since three lines of algebra based on the definition of correlation will give you the answer: Yes. It’s a common question in first year stats problem set. Even ignoring the weird decision to randomly generate L, which of course does absolutely nothing (since the correlation of X and Y is the same as the correlation of X/Land Y/L), the only thing this elaborate simulation you ran tells us is that you have less than rudimentary understanding of what correlation is.

2. Your conclusion is wrong:

Your statement that “The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker” is sloppy logic at best.

What the (silly, unnecessary) simulation shows is this:

The existence of a correlation between wages and value added cannot tell us which of two theories of wages is more correct: (a) the neo-classical theory (b) the theory that profits are determined randomly, and wages are also determined randomly, and that the two are independent of each other (in the statistical sense of independence).

Since both these theories share a prediction – that there will be a correlation between wages and value added – the realization of this prediction in the data cannot tell us anything about which theory of these two is more correct.

Of course, there is no theory that wages and profits are determined independently and randomly, so your simulation adds no useful information to the debate. Your use of “The reality is” is just wrong. The correlation is not because in “reality” the accounting definition virtually guarantees anything. The correlation you find is due to the way you assumed the relationship between the variables (namely, independently distributed).

If, “in reality”, wages are held at subsistence level because of the “army of unemployed” then there would be no correlation between wages and value added – even though the accounting is the same. In reality, something determine wages, and something determines profits. Assuming they are random and independent and calling that “in reality” is flat out wrong.

To sum up, your unnecessary simulation only shows that you can come up with another “theory” that predicts a correlation between wages and value added. This is a theory in which profits and wages are determined randomly and independently. I can come up with a hundred more before breakfast. This does not at all mean anything like “The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker.”

Show some humility. You clearly have very limited understanding of the issues, not even understanding what correlation is. There’s no shame in that – some of the smartest and most insightful people I know don’t understand any of these things. Just maybe you want to be a bit more cautious before declaring all of neo classical economics an obvious fraud.

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on second read, I see that you divide by a different L for each firm. This isn’t quite as meaningless, but still silly. What’s the point? It only means you draw the numbers from a different distribution, where of course the distribution itself does not matter. The correlation is basically determined from the definition of correlation and from your assumption of independence.

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Welcome back assafzim,

I was waiting for you to troll this post. As usual, you do not disappoint. Sadly, there’s very little to respond to here besides ad hominem attacks. It seems that you don’t understand how null hypotheses work. They are not “theories”. They are an assumption by which we judge correlations. We assume the variables are independent and randomly distributed. Then we see how likely it is to find the given correlation.

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you are wrong, and while I did let a couple of ad hominem slip, it’s only b/c I am truly amazed by the audacity: you trash a whole discipline as obviously flawed, trash a whole community of scholars as frauds, when you don’t even understand the basics. Anyways –

Minor point: it is incorrect that “we assume the variables are independent and randomly distributed”. While this is common, null hypothesis often assume different things. They may assume a perfect correlation, the may assume negative correlation, they may assume correlation of whatever magnitude. But that’s a minor issue.

Major point: what you do is NOT hypothesis testing.

Hypothesis testing would be to ask “if **wage and value added** were NOT in fact correlated, how likely is it that we would observe the correlation between **wage and value added** that we see in the data”.

This is clearly rejected.

Alternatively, hypothesis testing would be to ask if **wage and profits** were not in fact correlated, how likely is it that we would observe the correlation between **wage and profits** that we do in the data”. This too, btw, is clearly rejected.

What you do is assume **wage and profit** are independently distributed (and hence not correlated), and then ask how likely is it, given that **wage and profit** are not correlated, that we will find in the data that **wage and value added** are correlated. This is just sloppy thinking. It is certainly not hypothesis testing.

Also, it’s silly. If wage and profits are independent, then with three lines of first year algebra you can show that wage and value added are correlated. It’s just shocking that you would think this was somehow missed by generations of scholars.

And that, pointing out this trivial mathematical fact, is all your simulation does. It shows that if you assume that wages and value added are correlated (which is what you assume as it follows directly from your assumption that wages and profits are not correlated), and then generate a whole lotta samples of wages and value addeds, in the vast majority of them you will get that the sample of wages is correlated with the Sample of value added. Really. That is literally what your simulation does.

I am trying to give a coherent meaning to your point that the accounting makes it “virtually guarantee[d]” that a correlation between wages and value added will be found. It’s hard, but the only sense I can make of this statement is that “regardless of what is the actual process that determines wages and value added, the accounting virtually guarantees the data will show a correlation”. This is obviously false. Only because of your assumption that wage and profit are independent you find a correlation between wage and value added. If the process that determines wages and value added in reality was not such (and, btw, of course it is not such) then it is not at all guaranteed that there will be a correlation between wages and value added.

This is a good point to continue to the non-technical comment

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Here are things that would follow if your logic was correct:

1. Call a person expenditure on cigarettes W, and his expenditure on everything else P. His total expenditure is therefore W+P, which we can call Y. This virtually guarantees that expenditure on cigarettes is positively correlated with total expenditure.

But of course that is false. You can run this empirical test many times over, and in the vast majority of cases you will find a negative correlation.

2. Marx theory of wages, according to which wages are kept at subsistence, should have been empirically rejected without any need to actually do any empirical work. Since, by Marx theory, wages are not correlated with value added, and since the accounting of it all virtually guarantees we will find a correlation, there’s no need for empirical work to reject Marx theory.

Again – this is obviously a ridiculous statement.

3. I have a sample of only startups in Silicon Valley (zero value added as there are no sales yet, wages are just the negative of profits), and diamond mines in Africa (very low wages, very high value added). All the accounting is the same as in your post here. The accounting virtually guarantees that there is a positive correlation between wages and value added.

Again – this is false. In this example there is a negative correlation. Again – even though the accounting of wages, profits, and value added is the same as in your post.

ok, I think the point is clear. The accounting does not guarantee, not does it virtually guarantees, nor does it even imply that “in reality” there will be a correlation between wages and value added. The underlying process that generates the data is what determines the existence of the correlation or its absence. Therefore, the existence of the correlation or its absence does indeed teach us about the underlying process that generates the data.

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ok, I had time to organize my points, so let me offer the readers this clearer explanation of what this post is doing.

1. It uses a statistical software to generate many random values for wages and for profits. It defines wages and profits to be statistically uncorrelated with each other.

2. It then defines (correctly) added value as the sum of wages and profits. BY DEFINITION, a sum of two independent random variables (value added) is correlated with each of its components (wages, profits). So, by defining wages and profits to be independent, the post here defines value added to be correlated with wages.

3. The statistical software, obviously, works based on the instructions it is given. And it was given the instruction to generate wages and value added so that they are correlated.

4. Now, because there is some randomness in the process, not 100% of the samples the software will generate will indeed show a correlation. But of course – of course!!! – if the instructions to the statistical software were “please generate samples of wages and value added that are correlated”, which is exactly the instruction that it was given, the vast majority of these samples will indeed show a correlation.

5. Astonishingly, the post then argues that this exercise – telling a software to generate variables that are correlated, and then finding that they are indeed correlated in most cases – teaches us something about reality. It is saying that this exercise shows that “in reality” most samples of wages and value added have to be correlated, because of the accounting. This is just silly in the extreme.

6. The correlation in the samples that the statistical software created comes from – who would have thunk it – the fact that it was told to generate correlated variables, i.e. from the assumption that wages and profits are independent (which leads, mathematically, to the correlation between wages and value added).

7. In reality, what determines whether or not there is a correlation between wage and value added is the process that generates wages and value added. Even though they are related to each other by accounting, in reality, they may be positively correlated, uncorrelated, or negatively correlated. All depending on the process that generates the data. Numerous examples from reality show this very clearly.

8. Hypothesis testing, which seems to be the source of confusion in this post, deals with something entirely different. Hypothesis testing asks, for example, if X and Y in reality are not correlated in the whole population, how likely is it that we would see a correlation in a specific sample. It has absolutely nothing to do with what’s in this post.

9. Based on this sloppy thinking and misunderstanding of basic concepts, the author wants you to believe he has found a fatal flaw in decades of empirical research. sigh…

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